Feature adapted beamlet transform apparatus and associated methodology of detecting curvilinear objects of an image

ABSTRACT

A method of detecting a curvilinear object of a noisy image. The method includes filtering the noisy image in accordance with a two dimensional line profile. The line profile is selected within a class of steerable filters. A beamlet coefficient is calculated in accordance with the filtering, wherein a coefficient above a predetermined threshold identifies a local feature.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of the earlier filing date of U.S.Provisional Application No. 60/911,797, filed Apr. 13, 2007, entitled “AFeature Adapted Beamlet Transform Apparatus and Associated Methodologyof Detecting Curvilinear Objects of an Image” the entirety of which isincorporated herein by reference.

BACKGROUND

The claimed advancements described herein relate to a system andassociated methodology of detecting curvilinear objects in an image.More specifically, an apparatus and associated methodology are providedfor performing a feature adapted Beamlet transform for the detection ofcurvilinear objects in a noisy image via a software and/or hardwareimplementation.

In image processing systems computer vision applications and the like,the detection of curvilinear objects is often times desired. Suchobjects occur in every natural or synthetic image as contours ofobjects, roads in aerial linear imaging or DNA filaments in microscopyapplications. Currently, there is no known methodology in which asteerable filter may be leveraged to employ line segment processingmethodologies such as beamlet methods for representing curvilinearobjects carrying a specific line-profile.

Curvilinear objects are considered as 1 dimensional manifolds that havea specific profile running along a smooth curve. The shape of thisprofile may be an edge or a ridge-like feature. It can also berepresented by more complex designed features. For example, in thecontext of DNA filament analysis in fluorescent microscopy, it isacceptable to consider the transverse dimension of a filament to besmall relative to the PSF (point spread function) width of themicroscope. Hence, the shape of the profile may be accuratelyapproximate by a PSF model.

One way to detect curvilinear objects is to track locally the feature ofthe curve-profile; linear filtering or template matched filtering arewell-known techniques for doing so. Classical Canny edge detector andmore recently designed detectors are based on such linear filteringtechniques. They involve the computation of inner-products with shiftedand/or rotated versions of the feature template at every point in theimage. High response at a given position in the image means that theconsidered area has a similarity with the feature template. Filtering isusually followed by a non-maxima suppression and a thresholding step inorder to extract the objects. The major drawbacks of such approachescome from the fact that linear filtering is based on local operators.Hence it is highly sensitive to noise but not sensitive to theunderlying smoothness of the curve, which is a typical non-localproperty of curvilinear objects.

Alternatively, the Radon transform is a powerful non-local techniquewhich may be used for line detection. Also known as the Hough transformin the case of discrete binary images, it performs a mapping from theimage space into a line parameter space by computing line integrals.Formally, given an image f defined on a sub-space of R², for every lineparameter (p, θ), it computes

$\begin{matrix}{{\phi \left( {\rho,\vartheta} \right)}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{f\left( {x,y} \right)}{\delta \left( {\rho - {x\; {\cos (\theta)}} - {y\; {\sin (\theta)}}} \right)}\ {x}\ {{y}.}}}}} & (1)\end{matrix}$

Peaks in the parameter space reveals potential lines of interest. Thisis a very reliable method for detecting lines in noisy images. However,there are several limitations. First, direct extension of that method todetect more complex curves is unfeasible in practice for it increasesthe complexity exponentially by adding one dimension to the parameterspace. In addition, Radon transform computes line integrals on linesthat pass through the whole image domain and does not provideinformation on small line segments.

Given an image of N×N pixels, the number of possible line segmentsdefined is in O(N⁴). Direct evaluation of line integrals upon the wholeset of segments is practically infeasible due to the computationalburden. One of the methodologies proposed to address this problem is theBeamlet transform. It defines a set of dyadically organized linesegments occupying a range of dyadic locations and scales, and spanninga full range of orientations. This system of line segments, calledbeamlets, have both their end-points lying on dyadic squares that areobtained by recursive partitioning of the image domain. The collectionof beamlets has a O(N² log(N)) cardinality. The underlying idea of theBeamlet transform is to compute line integrals only on this smaller set,which is an efficient substitute of the entire set of segments for itcan approximate any segment by a finite chain of beamlets. Beamletchaining technique also provides an easy way to approximate piecewiseconstant curves.

Formally, given a beamlet b=(x, y, l, θ) centered at position (x,y),with a length l and an orientation θ, the coefficient of b computed bythe Beamlet transform is given by

$\begin{matrix}{{\Phi \left( {f,b} \right)} = {\int_{{- 1}/2}^{1/2}{{f\left( {{x + {{\gamma cos}(\theta)}},{y + {{\gamma sin}(\theta)}}} \right)}\ {{\gamma}.}}}} & (2)\end{matrix}$

Equation (2) is closely related to equation (1) since Beamlet transformcan be viewed as a multiscale Radon transform; they both integrate imageintensity along line segments. However, they do not take into accountany line-profile. It implies that the Radon and Beamlet transforms arenot well-adapted to represent curvilinear objects carrying a specificline-profile.

Accordingly, a feature-adapted Beamlet transform is provided torepresent curvilinear objects of a specific line profile.

SUMMARY OF EXEMPLARY ASPECTS OF THE ADVANCEMENTS

In one aspect, a method of detecting a curvilinear object of a noisyimage is provided. The method includes filtering the noisy image inaccordance with a two dimensional line profile. The line profile isselected within a class of steerable filters. A beamlet coefficient iscalculated in accordance with the filtering, wherein a coefficient abovea predetermined threshold identifies a local feature.

In a further aspect, a method of detecting a curvilinear object of anoisy image is disclosed. The method includes filtering the noisy imagein accordance with a two dimensional line profile. The line profile isselected within a class of steerable filters. A beamlet coefficient iscalculated in accordance with the filtering, wherein a coefficient abovea predetermined threshold identifies a local feature. The noisy image isconvolved in accordance with a number of basic filters.

In still a further aspect of the invention, a method of detecting acurvilinear object of a noisy image is disclosed. The method includesfiltering the noisy image in accordance with a two dimensional lineprofile. The line profile is selected within a class of steerablefilters. A beamlet coefficient is calculated in accordance with thefiltering, wherein a coefficient above a predetermined thresholdidentifies a local feature. The noisy image is convolved in accordancewith a number of basic filters and each image is computed by linearcombination.

It is to be understood that both the foregoing general description ofthe invention and the following detailed description are exemplary, butare not restrictive, of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fees.

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 illustrates a high level block diagram of a feature-adaptedBeamlet transform in accordance with an exemplary aspect of thedisclosure;

FIG. 2 illustrates an example of an original image before noisecorruption;

FIG. 3 illustrates the original image of FIG. 2 corrupted with noise;

FIG. 4 illustrates the use of 3^(rd) order edge detection of curvilinearobjects as applied to the noisy image in FIG. 3;

FIG. 5 illustrates detection using feature adapted beamlet transformcarrying the same 3^(rd) order filter, as applied to the noisy image inFIG. 3, in accordance with an exemplary aspect of the disclosure;

FIG. 6 illustrates an example or an original image of DNA filamentsobtained by fluorescent microscopy;

FIG. 7 illustrates the use of 2^(nd) order edge detection of curvilinearobjects using standard beamlet transform, as applied to the originalimage in FIG. 6; and

FIG. 8 illustrates detection using feature adapted beamlet transformcarrying the same 2^(nd) order filter, as applied to the original imagein FIG. 6, in accordance with an exemplary aspect of the disclosure.

DETAILED DESCRIPTION

A feature adapted beamlet transform apparatus and associated methodologyof detecting curvilinear objects of an image is provided to unify theBeamlet transform with a linear filtering technique to introduce theFeature-adpated Beamlet Transform, which is able to incorporateknowledge of a desired line-profile running along curves. If the profileis designed as a steerable filter, this methodology leads to anefficient implementation.

FIG. 1 shows the Feature-adapted Beamlet transform block diagram inaccordance with an exemplary embodiment. A typical input image 2 isshown for processing by the feature adapted Beamlet transform system,generally designated 5. The front end of the system includes a bank of adedicated basis filters 4, which convolve the image as input. The outputof the basis filters 4 is transformed with a Beamlet transform 6. Inthis regard, it is noted that the basis filters 4 and the Beamlettransform 6 are equally applicable to remotely distributed nodes, or,stand alone systems. For example, those skilled in the art willrecognize that such processing may be computed on independent systems.In such cases, the Beamlet transform may be employed by a stand alonesystem as a local utility for facilitating image recognition. As such,it is to be understood that basis filters 4 and the Beamlet 6 transformmay correspond to separate devices or separate aspects of a same devicein accordance with the advancements described herein. Likewise,components 6 and 8, although illustrated as separate objects, are inexemplary embodiments described herein, components of a feature adaptedBeamlet transform apparatus.

The output from the Beamlet transform 6 is multiplied by a set of gainmaps 8, which apply the appropriate interpolation functions at eachposition and time. The summation junction 10 produces the adaptivelyfiltered and transformed image 12.

The system 5 of FIG. 1 may embrace a personal computing device such as aPC employing an Intel Pentium processor. The instruction set describedin detail below is provided as a utility application executing inconjunction with a local processor and operating system such asMicrosoft VISTA®, Unix, Solaris, Linux, Apple MAC-OS and other systemsknown to those skilled in the art. Alternatively, those skilled in theart will recognize the applicability to mobile devices such as PDAs,phones, and portable entertainment devices which employ Symbian,Microsoft Mobile® and like operating systems. Memory required forsupporting the registries, kernel and like features of FIG. 1 is omittedas well known. Likewise the description of general features of FIG. 1such as local volatile and/or non-volatile memory, I/O capabilities,common peripheral devices, as well as corresponding functionality havebeen omitted for brevity, the detailed operation/description of which iswell known to those skilled in the art. The specific coding and portingof the algorithms described herein is within the ability of one skilledin the art upon review of this specification.

It is understood that within the scope of the appended claims, theinventions may be practiced otherwise than as specifically describedherein. For example, while described in terms of both software andhardware components interactively cooperating, it is contemplated thatthe feature adapted beamlet transform described herein may be practicedentirely in software, firmware, or as an Application Specific IntegratedCircuit (ASIC).

As recognized by those skilled in the art, software and firmware may beembodied on a computer readable storage medium such as an optical discor semiconductor memory.

Moreover, the feature adapted beamlet transform may be implemented as aweb based utility or a web based service invoked remotely by a knownprotocol such as SOAP. For example, it is envisioned that the featureadapted beamlet transform may be leveraged in a research environmentwhere images are provided to the utility via a network for servicing agroup of users. Likewise, remote devices may be employed to access thefeature adapted beamlet transform via a number of wireless protocolssuch as BLUETOOTH® and I.E.E.E.802-11x wireless formats.

All steps relative to a single basis filter can be simultaneouslycomputed on a parallel machine. All these steps have a O(N²) complexity.In this scheme, the evaluation of beamlet coefficients consumes most ofthe computation time. To increase speed a cache strategy may be employedto pre-compute most of the computation while utilizing an approximationof beamlet coefficients based on the two-scale recursion technique. Thisarrangement increases speed at the expense of a memory load. For a1024×1024 image, an implementation of the standard Beamlet transformtakes approximately 1 s on a dual processors based computer.

Consider a filter h representing a 2-dimensional line-profile. Let h^(θ)be a rotated version of h in the direction θ:

h ^(θ)(x, y)=h(R _(θ)(x, y)),  (3)

where R_(θ) is the 2-dimensional rotation matrix of angle θ. In a firststep, consider filtering image f 2 with h^(θ) before computing thebeamlet coefficient from equation (2):

$\begin{matrix}{{\Phi \left( {f,b} \right)} = {\int_{{- 1}/2}^{1/2}{{f\left( {{x + {{\gamma cos}(\theta)}},{y + {{\gamma sin}(\theta)}}} \right)}\ {{\gamma}.}}}} & (2)\end{matrix}$

This yields:

$\begin{matrix}{{\Psi \left( {f,b} \right)} = {\int_{{- 1}/2}^{1/2}{f*{h^{\theta}\left( {{x + {{\gamma cos}(\theta)}},{y + {{\gamma sin}(\theta)}}} \right)}\ {{\gamma}.}}}} & (4)\end{matrix}$

A high coefficient identifies that the local feature runs significantlyalong b. This is the Feature-adapted Beamlet transform 6. In general,the computation of all beamlet coefficients is not conceivable, since itrequires to convolve the image as many times as the number of θ's. Forthe special case where h is selected to be within the class of steerablefilters, consider writing h^(θ) as a linear combination of basis filters4:

$\begin{matrix}{{{h^{\theta}\left( {x,y} \right)} = {\sum\limits_{j = 1}^{M}{{k_{j}(\theta)}{h^{\theta_{j}}\left( {x,y} \right)}}}},} & (5)\end{matrix}$

where k_(j)'s 8 are interpolation functions that only depend on θ. Thebasis filters h^(θ) ^(j) 's 4 are independent of θ. A convolution of animage with a steerable filter of arbitrary orientation is then equal toa finite weighted sum of convolution of the same image with the basisfilters. Hence, equation (4) can be written as

$\begin{matrix}\begin{matrix}{{\Psi \left( {f,b} \right)} = {\sum\limits_{j = 1}^{M}{{k_{j}(\theta)}{\int_{{- 1}/2}^{1/2}{{f^{\theta_{j}}\left( {{x + {{\gamma cos}(\theta)}},{y + {{\gamma sin}(\theta)}}} \right)}\ {\gamma}}}}}} \\{{= {\sum\limits_{j = 1}^{M}{{k_{j}(\theta)}{\Phi \left( {f^{\theta_{j}},b} \right)}}}},}\end{matrix} & (6)\end{matrix}$

where f^(θ) ^(j) =f*h^(θ) ^(j) and Φ(f^(θ) ^(j) ,b) corresponds to thebeamlet coefficient of b computed over f^(θ) ^(j) using equation (2). Asa result, in order to compute equation (4) for every beamletcoefficient, consider the following: first convolve the image as manytimes as the number of basis filters composing our filter h. This numberis typically very small. On each filtered image, compute the standardBeamlet transform. Finally, for each beamlet, compute its coefficientusing equation (6).

A detection method using the Feature-adpated Beamlet transform 10provides a list of beamlets that best represent curvilinear objectscarrying a specific line-profile in an image. The method is based on amultiscale coefficient thresholding technique.

A Recursive Dyadic Partition (RDP) of the image domain is any partition,starting from the whole image domain, obtained by recursively choosingbetween replacing any square of the partition by its decomposition intofour dyadic squares or leaving it unsplit. This concept is very similarto the quadtree decomposition technique. A beamlet-decorated RDP(BD-RDP) is a RDP in which terminal nodes of the partition areassociated with at most one beamlet. By construction, BD-RDP provides alist of non-overlapping beamlets. In order to select the list ofbeamlets that best represent curvilinear objects in the image 2,maximize over all beamlet-decorated recursive dyadic partitions P={S₁,S₂, . . . , S_(n)} the following complexity penalized residual sum ofsquare:

$\begin{matrix}{{{{E(P)} = {{\sum\limits_{S \in P}C_{S}^{2}} - {\lambda^{2}\# P}}},{where}}{C_{s} = {\max\limits_{b \in S}\frac{{FBT}\left( {f,b} \right)}{\sqrt{l}}}}} & (7)\end{matrix}$

measures the energy required to model the region S of the image f by thebeamlet b and λ is a MDL-like criteria that controls the complexity ofthe model. A high value of λ yields to a coarse representation ofcurvilinear structures; a small value leads to a quite complex modelwith potentially a significant number of false alarms. Equation (7) canbe solved very efficiently by a recursive tree-pruning algorithm due toadditivity of the cost function.

Consider comparing this methodology with a linear filtering techniquewhich convolves the image with a steerable filter and resolves for eachimage point a polynomial equation in order to find the optimalorientation maximizing the filter response. This step is followed by anon-maxima suppression and a thresholding step. Exemplary steerablefilters are a combination of Gaussian-based filters which are optimizedunder Canny-like criteria, such as a 3^(rd) order filter. FIG. 3 showsresults on a noisy image corrupted by Gaussian white noise with standarddeviation σ_(noise)=50. In the methods described herein, the well-knownBresenham algorithm is used to highlight pixels traversed by meaningfulbeamlets. In both cases, the threshold value is determined to keep 2,000pixels. As shown in FIG. 5, the number of false positives is highlyreduced.

Consider evaluating the performance of the Feature-adapted Beamlettransform 10 compared to the standard Beamlet transform for thedetection of multiple lines segments in noisy images. Two techniques aretested on images of DNA filaments obtained by fluorescent microscopy.These filaments have a ridge-like profile. For the choice of h, wechoose a 2^(nd) order filter. The same algorithm is used for bothtransforms with λ=100. The standard Beamlet transform behaves like alow-pass filter and hence, is sensitive to the background intensity, asopposed to the Feature-adpated Beamlet transform 10 which can cancelconstant or more complex background, depending on the vanishing momentsof h. In the example shown herein, in order to get these two transformscomparable between each other, the background is assumed to be constantand is substracted from the image before computing the beamletcoefficients. To do so, the background mean intensity is estimated fromthe median of the image 2. As shown in the top left corner of FIG. 6,the spurious detections are due to the fact that real background is notconstant over the whole image domain. As can be seen in FIG. 7, this isnot the case for the exemplary feature adapted Beamlet transform.

Thus, the foregoing discussion discloses and describes merely exemplaryembodiment of the present invention. As will be understood by thoseskilled in the art, the present invention may be embodied in otherspecific forms without departing from the spirit or essentialcharacteristics thereof. Accordingly, the disclosure of the presentinvention is intended to be illustrative, but not limiting of the scopeof the invention, as well as other claims. The disclosure, including anyreadily discernible variants of the teachings herein, define, in part,the scope of the foregoing claim terminology.

1. A method of detecting a curvilinear object of a noisy image,comprising: filtering the noisy image in accordance with a twodimensional line profile, the line profile being selected within a classof steerable filters; and calculating a beamlet coefficient inaccordance with the filtering wherein a coefficient above apredetermined threshold identifies a local feature.
 2. The methodaccording to claim 1 further comprising: convolving a noisy image inaccordance with a number of basis filters.
 3. The method according toclaim 2 wherein a filtered image is computed with a beamlet transformapplication.
 4. The method according to claim 3 wherein each image iscomputed by linear combination.
 5. A computer readable storage mediumincluding encoded computer program instructions that causes a computerto detect a curvilinear object of a noisy image, comprising: filteringthe noisy image in accordance with a two dimensional line-profile, theline profile being selected within a class of steerable filers; andcalculating a beamlet coefficient in accordance with the filteringwherein a coefficient above a predetermined threshold identifies a localfeature.
 6. The medium according to claim 5 further comprising:convolving a noisy image in accordance with a number of basis filters.7. The medium according to claim 5 wherein a filtered image is computedwith a beamlet transform application.
 8. The medium according to claim 5wherein each image is computed by linear combination.